Properties

Label 1380.107
Modulus $1380$
Conductor $1380$
Order $44$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1380, base_ring=CyclotomicField(44))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([22,22,11,34]))
 
pari: [g,chi] = znchar(Mod(107,1380))
 

Basic properties

Modulus: \(1380\)
Conductor: \(1380\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1380.bv

\(\chi_{1380}(83,\cdot)\) \(\chi_{1380}(107,\cdot)\) \(\chi_{1380}(143,\cdot)\) \(\chi_{1380}(203,\cdot)\) \(\chi_{1380}(227,\cdot)\) \(\chi_{1380}(263,\cdot)\) \(\chi_{1380}(287,\cdot)\) \(\chi_{1380}(383,\cdot)\) \(\chi_{1380}(467,\cdot)\) \(\chi_{1380}(503,\cdot)\) \(\chi_{1380}(527,\cdot)\) \(\chi_{1380}(563,\cdot)\) \(\chi_{1380}(707,\cdot)\) \(\chi_{1380}(743,\cdot)\) \(\chi_{1380}(803,\cdot)\) \(\chi_{1380}(983,\cdot)\) \(\chi_{1380}(1187,\cdot)\) \(\chi_{1380}(1247,\cdot)\) \(\chi_{1380}(1307,\cdot)\) \(\chi_{1380}(1367,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{44})\)
Fixed field: Number field defined by a degree 44 polynomial

Values on generators

\((691,461,277,1201)\) → \((-1,-1,i,e\left(\frac{17}{22}\right))\)

Values

\(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(29\)\(31\)\(37\)\(41\)\(43\)
\(1\)\(1\)\(e\left(\frac{19}{44}\right)\)\(e\left(\frac{21}{22}\right)\)\(e\left(\frac{25}{44}\right)\)\(e\left(\frac{7}{44}\right)\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{21}{44}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{5}{44}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1380 }(107,a) \;\) at \(\;a = \) e.g. 2